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A Short Introduction to Topological Quantum Computation

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A Short Introduction to Topological Quantum Computation SciPost Phys. 3, 021 (2017) A short introduction to topological quantum computation Ville T. Lahtinen1 and Jiannis K. Pachos2? 1 Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany 2 School of Physics and Astronomy, University of Leeds, Leeds, LS2 9JT, United Kingdom ? [email protected] Abstract This review presents an entry-level introduction to topological quantum computation – quantum computing with anyons. We introduce anyons at the system-independent level of anyon models and discuss the key concepts of protected fusion spaces and statisti- cal quantum evolutions for encoding and processing quantum information. Both the encoding and the processing are inherently resilient against errors due to their topolog- ical nature, thus promising to overcome one of the main obstacles for the realisation of quantum computers. We outline the general steps of topological quantum computa- tion, as well as discuss various challenges faced by it. We also review the literature on condensed matter systems where anyons can emerge. Finally, the appearance of anyons and employing them for quantum computation is demonstrated in the context of a sim- ple microscopic model – the topological superconducting nanowire – that describes the low-energy physics of several experimentally relevant settings. This model supports lo- calised Majorana zero modes that are the simplest and the experimentally most tractable types of anyons that are needed to perform topological quantum computation. Copyright V.T. Lahtinen and J. K. Pachos. Received 12-05-2017 This work is licensed under the Creative Commons Accepted 24-08-2017 Check for Attribution 4.0 International License. Published 09-09-2017 updates Published by the SciPost Foundation. doi:10.21468/SciPostPhys.3.3.021 Contents 1 Introduction2 1.1 Topology, stability and anyons3 2 Topological order and anyons in condensed matter systems6 2.1 Topological states that support anyons7 2.2 Manifestations of anyons in microscopic many-body systems9 2.2.1 Degeneracy and Berry phases9 2.2.2 Topological degeneracy and entanglement entropy 11 3 Anyon models 12 3.1 Fusion channels - Decoherence-free subspaces 13 3.2 Braiding anyons - Statistical quantum evolutions 15 3.3 Example 1: Fibonacci anyons 15 3.4 Example 2: Ising anyons 17 1 SciPost Phys. 3, 021 (2017) 4 Quantum computation with anyons 19 4.1 Initialization of a topological quantum computer 19 4.2 Quantum gates – Braiding anyons 20 4.3 Measurements – Fusing anyons 21 4.4 Possible error sources 22 5 Topological quantum computation with superconducting nanowires 24 5.1 Majorana zero modes in a superconducting nanowire 25 5.2 The Majorana qubit 27 5.3 Manipulating and reading out the Majorana qubit 29 5.4 Challenges with Majorana-based topological quantum computation 30 6 Outlook 30 References 31 1 Introduction Topological quantum computation is an approach to storing and manipulating quantum infor- mation that employs exotic quasiparticles, called anyons. Anyons are interesting on their own right in fundamental physics, as they generalise the statistics of the commonly known bosons and fermions. Due to this exotic statistical behaviour, they exhibit non-trivial quantum evo- lutions that are described by topology, i.e. they are abstracted from local geometrical details. When anyons are used to encode and process quantum information, this topological behaviour provides a much desired resilience against control errors and perturbations. To be more pre- cise, the presence of certain kinds of anyons gives rise to a degenerate decoherence-free sub- space, in which the state can only be evolved by moving the anyons adiabatically around each other. While from the first sight anyons appear to be an over-complicated method for per- forming quantum computation, they are profoundly linked to quantum error correction [1], the algorithmic means we have in dealing with errors during quantum computation. In a sense, anyonic quantum computers implement quantum error-correction at the hardware level, thus becoming resilient to control errors and erroneous perturbations. This has augmented topo- logical quantum computation from a niche field of research to a methodology that permeates much of the research efforts in realising fault-tolerant quantum computation. In this review we present a non-technical introduction to anyons and to the framework for performing fault-tolerant quantum computation with them. The emphasis will be on the key properties that define anyons and how they enable protected encoding and processing of quantum information. As anyons can emerge in numerous microscopically distinct systems, we discuss these concepts primarily at a platform-independent level of anyon models and provide an extensive list of references for the interested reader to go deeper. Our aim is to provide a clear and concise introduction to the underlying principles of topological quantum computation without expertise in condensed matter theory. When condensed matter concepts are needed, we introduce them in a heuristic level to give the reader a general understanding without referring to the mathematical details. In doing so, we aim this review to be accessible to anyone with a solid undergraduate understanding of quantum mechanics and the basics of quantum information. While several reviews have already been written on the topic, we aim this review to serve as 2 SciPost Phys. 3, 021 (2017) an accessible starting point. Topological quantum computation with fractional quantum Hall States is reviewed extensively by Nayak et al. [2], while Das Sarma et al. focus on quantum computation with Majorana zero modes [3]. The book by Pachos can be viewed as an extended version of the present review that goes deeper into the condensed matter topics [4], while the book by Wang focuses on the more mathematical aspects and their connections to knot theory [5]. The reader may also find useful the lecture notes by Roy and DiVincenzo [6] and the classic lecture notes by Preskill [7]. This review concerns only topological quantum computation where both the encoding and processing of quantum information is topologically protected. Anyons have applications also to quantum memories and quantum error correction, i.e. when only the encoding is topologically protected. For reviews on topological quantum memories, we refer the interested reader to the reviews by Terhal [8] and Brown et al. [9]. This review is structured as follows. We begin in Section 1.1 by describing at a heuristic level why topology can increase fault-tolerance and why the dimension of space is paramount when looking for systems that support anyons. In Section2 we discuss the different types of topological order and the conditions under which they can support anyons. An extensive list of known systems of anyons is provided and we also outline how the defining properties of anyons manifest themselves in microscopic systems. In Section3 we turn to the system independent discussion of anyon models and describe how a minimal set of data captures all the dynamics associated with a given anyon model. As examples we consider both Fibonacci anyons (what we would like to have for topological quantum computation) and Ising anyons (what we have so far). Section4 is the core of the review where we explicitly discuss how Ising anyons can be used to encode and process quantum information in a topologically protected manner, while in Section5 we illustrate how such quantum computation could be carried out in a specific microscopic system. As an example we employ superconducting nanowire arrays that support Majorana zero modes and that are currently the experimentally most promising direction. We conclude with Section6. 1.1 Topology, stability and anyons In mathematics, topology is the study of the global properties of manifolds that are insensitive to local smooth deformations. The overused, but still illustrative example is the topological equivalence between a donut and a coffee cup. Regardless of the local details that give them rather different everyday practicalities, both are mathematically described by genus one man- ifolds meaning that there is a single hole in both. Small smooth deformations, such as taking a bite on the side of the donut or chipping away a piece of the cup will change the object locally, but the topology remains unchanged. Only global violent deformations, such as cutting the donut in half or breaking the cup handle, will change the topology by removing the hole. However, in real world small deformations matter. Quite spectacular salesmanship is re- quired to sell a donut from which someone has already taken a bite. Something similar occurs also in quantum mechanics. To store and evolve a pure quantum state coherently, one must take exceptional care that no outside noise interferes and that the evolution is precisely the desired one. This is the key fundamental challenge in quantum computation: to robustly store quantum states for long times and evolve them according to specific quantum gates. Were quantum information encoded in topological properties of matter, and were the quan- tum gates dependent only on the topology of the evolutions, then both should be inherently protected from local perturbations. Such topological quantum computation would exhibit in- herent hardware-level stability that ideally would make elaborate schemes of quantum error- correction redundant. This idea was first floated by Kitaev in connection to surface codes for quantum error correction [1,10]. He realized that certain codes could be viewed as spin lattice models, where the elementary excitations are anyons – quasiparticles with statistics interpolating between 3 SciPost Phys. 3, 021 (2017) � � � � 3D 2D Figure 1: Exchange statistics in 2D vs. 3D. In 3D the path λ2 describing two particle exchanges is continuously deformable to λ1 by taking it behind or front of the right-most particle, and in turn λ1 is contractible to a point. Hence, all the paths have the same topology and thus correspond to the same statistical quantum evolution. In 2D, however, the paths λ2 and λ1 are topologically inequivalent since λ2 can not be deformed through the right-most particle, while λ1 is still contractible to a point.
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